Mathematics Commons^™

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov-Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schrodinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Integrable Models For Shallow Water With Energy Dependent Spectral Problems, Rossen Ivanov, Tony Lyons

Articles

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse problem as a Riemann–Hilbert problem with a Z₂ reduction group. The soliton solutions are explicitly obtained.

On Projection-Invariant Subgroups Of Abelian P-Groups, Brendan Goldsmith

Articles

A subgroup P of an Abelian p-group G is said to be projection-invariant in G if Pf is contained in P for all idempotent endomorphisms f. Clearly fully invariant subgroups are projection invariant, but the converse is not true in general. Hausen and Megibben have shown that in many familiar situations these two concepts coincide. In a different direction, the authors have previously introduced the notions of socle-regular and strongly socle-regular groups by focussing on the socles of fully invariant and characteristic subgroups of p-groups. In the present work the authors examine the socles of projection-invariant subgroups of Abelian p-groups.

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov–Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schr¨odinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Abstract Criteria For Multiple Solutions To Nonlinear Coupled Equations Involving Magnetic Schrodinger Operators, Mattias Enstedt, Michael Melgaard

Articles

We consider a system of nonlinear coupled equations involving magnetic Schrodinger

operators and general potentials. We provide a criteria for the existence of multiple

solutions to these equations. As special cases we get the classical results on

existence of innitely many distinct solutions within Hartree and Hartree-Fock

theory of atoms and molecules subject to an external magnetic fields. We also

extend recent results within this theory, including Coulomb system with a constant

magnetic field, a decreasing magnetic field and a "physically measurable" magnetic field.

On The N-Wave Equations And Soliton Interactions In Two And Three Dimensions, Vladimir S. Gerdjikov, Rossen Ivanov, Assen V. Kyuldjiev

Articles

Several important examples of the N-wave equations are studied. These integrable equations can be linearized by formulation of the inverse scattering as a local Riemann–Hilbert problem (RHP). Several nontrivial reductions are presented. Such reductions can be applied to the generic N-wave equations but mainly the 3- and 4-wave interactions are presented as examples. Their one and two-soliton solutions are derived and their soliton interactions are analyzed. It is shown that additional reductions may lead to new types of soliton solutions. In particular the 4-wave _{equations with} Z₂xZ₂ reduction group allow breather-like solitons. Finally it is demonstrated that …

Completeness Of Interacting Particles, Pavel Abramski

Doctoral

This thesis concerns the completeness of scattering states of n _-interacting particles in one dimension. Only the repulsive case is treated, where thereare no bound states and the spectrum is entirely absolutely continuous, so the scattering Hilbert space is the whole of L2(Rn). The thesis consists of 4 chapters: The first chapter describes the model, the scattering states as given by the Bethe Ansatz, and the main completeness problem. The second chapter contains the proof of the completeness relation in the case of two particles: n = 2. This case had in fact already been treated by B. Smit (1997), …

Euler Equations On A Semi-Direct Product Of The Diffeomorphisms Group By Itself, Joachim Escher, Rossen Ivanov, Boris Kolev

Articles

The geodesic equations of a class of right invariant metrics on the semi-direct product of two Diff(S) groups are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra are found.

Solutions To Quasi-Relativistic Multi-Configurative Hartree-Fock Equations In Quantum Chemistry, Carlos Argáez García, Michael Melgaard

Articles

We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N-electron Coulomb systems with quasi-relativistic kinetic energy for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge. The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.

Mathematical Methods For Biosensor Models, Qi Wang

Doctoral

A biosensor is defined as a compact analytical device incorporating a biological sensing element integrated within a physico-chemical transducer whose aim is to produce optical or electronic signals proportional to the concentration of an analyte in a sample. Biosensors offer enormous potential to detect a wide range of analytes in health care, the food industry, environmental monitoring, security and defence. The beneficial impact on society as a result of the availability of such systems is immense, therefore investigating any strategy that could reduce development times and costs and reveal alternative designs is of utmost importance. In particular, mathematical modelling and …

On The Acceleration Of Explicit Finite Difference Methods For Option Pricing, Stephen O'Sullivan, Conall O'Sullivan

Articles

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant–Friedrichs–Lewy (CFL) condition which limits the latter class’s efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can …

Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov

Articles

The relations between smooth and peaked soliton solutions are reviewed for the Camassa- Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann- Hilbert problem. The momentum map from the action-angle scattering variables T^∗(T^N) to the flow momentum (X^∗) provides the Eulerian representation of the N-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The …

A Stochastic Model For Wind Turbine Power Quality Using A Levy Index Analysis Of Wind Velocity Data, Jonathan Blackledge, Eugene Coyle, Derek Kearney

Conference papers

The power quality of a wind turbine is determined by many factors but time-dependent variation in the wind velocity are arguably the most important. After a brief review of the statistics of typical wind speed data, a non- Gaussian model for the wind velocity is introduced that is based on a Levy distribution. It is shown how this distribution can be used to derive a stochastic fractional diusion equation for the wind velocity as a function of time whose solution is characterised by the Levy index. A Levy index numerical analysis is then performed on wind velocity data for both …

A Stability Study Of A New Explicit Numerical Scheme For A System Of Differential Equations With A Large Skew-Symmetric Component, Katharine Gurski, Stephen O'Sullivan

Articles

Abstract. Explicit numerical methods for the solution of a system of stiff differential equations suffer from a time step size that approaches zero in order to satisfy stability conditions. Implicit schemes allow a larger time-step, but require more computations. When the differential equations are dominated by a skew-symmetric component, the problem is not stiffness in the sense that the size of the eigenvalues are unequal, rather the that the real eigenvalues are dominated by imaginary eigenvalues. We present and compare analytical results for stable time step limits for several explicit methods including the super-time-stepping method of Alexiades, Amiez, and Gremaud …

Daemon Decay And Cosmic Inflation, Emil Prodanov

Articles

Quantum tunneling in Reissner--Nordstroem geometry is studied and the tunneling rate is determined. A possible scenario for cosmic inflation, followed by reheating phases and subsequent radiation-domination expansion, is proposed.

On The Singular Weyl-Titchmarsh Function Of Perturbed Spherical Schrödinger Operators, Aleksey Kostenko, Gerald Teschl

Articles

We investigate the singular Weyl–Titchmarsh m-function of perturbed spherical Schrödinger operators (also known as Bessel operators) under the assumption that the perturbation q(x) satisfies xq(x) ∈ L1(0, 1). We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular mfunction belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.

On The (Non)-Integrability Of Kdv Hierarchy With Self-Consistent Sources, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov

Articles

Nonholonomic deformations of integrable equations of the KdV hierarchy are studied by using the expansions over the so-called “squared solutions” (squared eigenfunctions). Such deformations are equivalent to a perturbed model with external (self-consistent) sources. In this regard, the KdV6 equation is viewed as a special perturbation of KdV. Applying expansions over the symplectic basis of squared eigenfunctions, the integrability properties of the KdV6 equation are analysed. This allows for a formulation of conditions on the perturbation terms that preserve its integrability. The perturbation corrections to the scattering data and to the corresponding action-angle (canonical) variables are studied. The analysis shows …

Rational Bundles And Recursion Operators For Integrable Equations On A.Iii-Type Symmetric Spaces, Vladimir Gerdjikov, Georgi Grahovski, Alexander Mikhailov, Tihomir Valtchev

Articles

We analyze and compare the methods of construction of the recursion operators for a special class of integrable nonlinear differential equations related to A.III-type symmetric spaces in Cartan’s classification and having additional reductions.

Polynomial Bundles And Generalised Fourier Transforms For Integrable Equations On A.Iii-Type Symmetric Spaces, Vladimir Gerdjikov, Georgi Grahovski, Alexander V. Mikhailov, Tihomir Valchev

Articles

A certain class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions is analysed via the Inverse Scattering Method (ISM). The class contains systems of nonlinear evolution equations (NLEEs) associated with a Lax operator (for the time-evolution) polynomial in the spectral parameter. Using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the `squared solutions' (generalised exponentials) are derived. Next, expansions of Q and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform …

On Socle-Regularity And Some Notions Of Transitivity For Abelian P-Groups, Brendan Goldsmith, Peter Danchev

Articles

In the present work the interconnections between various notions of transitivity for Abelian p-groups and the recently introduced concepts of socle-regular and strongly socle-regular groups are studied.

Bayesian Inference For Exponential Random Graph Models, Alberto Caimo, Nial Friel

Articles

Exponential random graph models are extremely difficult models to handle from a statistical viewpoint, since their normalising constant, which depends on model parameters, is available only in very trivial cases. We show how inference can be carried out in a Bayesian framework using a MCMC algorithm, which circumvents the need to calculate the normalising constants. We use a population MCMC approach which accelerates convergence and improves mixing of the Markov chain. This approach improves performance with respect to the Monte Carlo maximum likelihood method of Geyer and Thompson (1992).

On Adjoint Entropy Of Abelian Groups, Brendan Goldsmith, Ketao Gong

Articles

The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. In the present work we introduce a \lq dual\rq \ notion based upon the replacement of the finite groups used in the definition of algebraic entropy, by subgroups of finite index. The basic properties of this new entropy are established and a connection to Hopfian groups is investigated.

Holomorphic Liftings From Infinite Dimensional Spaces, Sean Dineen, Milena Venkova

Articles

We obtain a number of positive solutions to a holomorphic lifting problem on a domain in a locally convex space.

Extended Green-Liouville Asymptotics And Vacuum Polarization For Lukewarm Black Holes, Cormac Breen, Adrian Ottewill

Articles

We consider a quantum field on a lukewarm black hole spacetime. We introduce a new uniform approximation to the radial equation, constructed using an extension of Green-Liouville asymptotics. We then use this new approximation to construct the renormalized vacuum polarization in the Hartle-Hawking vacuum. Previous calculations of the vacuum polarization rely on the WKB approximation to the solutions of the radial equation, however the nonuniformity of the WKB approximations obscures the results of these calculations near both horizons. The use of our new approximation eliminates these obscurities, enabling us to obtain explicitly finite and easily calculable values of the vacuum …

An Explicit Super‐Time‐Stepping Scheme For Non‐Symmetric Parabolic Problems, Stephen O'Sullivan, Katharine Gurski

Conference papers

Explicit numerical methods for the solution of a system of differential equations may suffer from a time step size that approaches zero in order to satisfy stability conditions. When the differential equations are dominated by a skew-symmetric component, the problem is that the real eigenvalues are dominated by imaginary eigenvalues. We compare results for stable time step limits for the super-time-stepping method of Alexiades, Amiez, and Gremaud (super-time-stepping methods belong to the Runge-Kutta-Chebyshev class) and a new method modeled on a predictor-corrector scheme with multiplicative operator splitting. This new explicit method increases stability of the original super-time-stepping whenever the skew-symmetric …

Inverse Scattering Transform For The Degasperis–Procesi Equation, Adrian Constantin, Rossen Ivanov, Jonatan Lenells

Articles

We develop the Inverse Scattering Transform (IST) method for the Degasperis- Procesi equation. The spectral problem is an sl(3) Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method are presented.

The Maximal Pure Spectrum Of An Abelian Group, Brendan Goldsmith, Ruediger Goebel

Articles

This paper introduces the notion of the maximal pure spectrum of an Abelian group - this is the set of isomorphism classes of maximal proper pure subgroups - and focuses on the situation in which this spectrum is small. The converse situation is also examined i.e. given a collection of isomorphism classes of groups, can one find an Abelian group having precisely this collection as its maximal pure spectrum. Finally, it is shown that in some familiar situations, the answers to these questions may be undecidable.

From Permutahedron To Aassociahedron, Colum Watt, Thomas Brady

Articles

For each finite real reflection group $W$, we identify a copy of the type-$W$ simplicial generalised associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalised associahedron and the elements of the type $W$ non-crossing partition lattice which is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for $W$.

The Similarity Problem For Indefinite Sturm-Liouville Operators With Periodic Coefficients, Aleksey Kostenko

Articles

We investigate the problem of similarity to a self-adjoint operator for $J$-positive Sturm-Liouville operators $L=\frac{1}{\omega}(-\frac{d^2}{dx^2}+q)$ with $2\pi$-periodic coefficients $q$ and $\omega$. It is shown that if 0 is a critical point of the operator $L$, then it is a singular critical point. This gives us a new class of $J$-positive differential operators with the singular critical point 0. Also, we extend the Beals and Parfenov regularity conditions for the critical point $\infty$ to the case of operators with periodic coefficients.

Climbing Eements In Finite Coxeter Groups, Colum Watt, Thomas Brady, Aisling Kenny

Articles

We define the notion of a climbing element in a finite real reflection group relative to a total order on the reflection set and we characterise these elements in the case where the total order arises from a bipartite Coxeter element.